The orbit you describe is typical for (moderately) long-period comets.
Given the desired orbital period, we can calculate the semi-major axis of the orbit (which, for a highly elliptical orbit like this, is approximately half its maximum distance from the star) using the formula $$T^2 = \frac{4\pi^2}{GM}a^3,$$ where $T$ is the orbital period, $a$ is the semi-major axis, $M$ is the mass of the star (the mass of the planet is assumed to be negligible in comparison) and $G$ is the universal gravitational constant. However, if we assume that the planet is orbiting a star of about the same mass as our Sun, and if we measure time in years and distance in AUs, we can take a shortcut: we know the Earth orbits the Sun in one year, with a semi-major axis of one AU, so we must have $a=1$ when $T=1$. The constant of proportionality thus simply becomes $1$, and we're left with $a = T^{2/3}$. For $T=1027$ (years), this works out to $a\approx 101.8$ (AU), so our hypothetical planet gets no further than $2a\approx203.6$ AU from the Sun.
This is beyond the classical Kuiper belt, but much closer to the Sun than the Oort cloud, which is believed to be the source of most long-period comets. It is also closer to the Sun than several recently discovered "detached objects" such as 90377 Sedna, which are believed to have relatively stable orbits. Your planet thus remains well within the Sun's gravitational well, and is at little risk of being scattered by passing stars. (Well, at least as long as they don't pass too close, in which case we could all be in trouble anyway.)
There is a problem, though.
Your planet might be safe and stable as long as it's far away from the Sun, but as soon as its orbit takes it into the inner solar system, it's going to occasionally pass close to to the planets already there, and thus be affected by their gravity. Such interactions could have many possible outcomes, but the statistically most likely one (at least in a solar system similar to ours) is a close pass with a large gas planet like Jupiter, which will either fling the "comet planet" straight out of the system, or, more likely, kick it into a shorter-period orbit that will keep it interacting with the other planets in the system until it gets ejected from the solar system, collides with the Sun or one of the planets, or (least likely) get captured into a stable near-circular orbit.
So, to keep our hypothetical planet safe, we really should clear the solar system of any inner planets that might disturb its orbit. Fortunately, that might not be so implausible — something obviously caused our planet to end up in that highly elliptical orbit in the first place, and it's not altogether implausible that the same something (which could've been, say, another star passing through the inner solar system) might have also conveniently swept the system clean of any inner planets in more conventional orbits. (Indeed, our planet might well have been one of those, before it was scattered into its new eccentric orbit.)
OK, so we've got the orbit stabilized. What about the climate?
First of all, let me note that there's absolutely no reason to expect a planet with an orbit like this to be tidally locked, and certainly not to a 1:1 resonance like you suggest. On such an eccentric orbit, the planet would feel almost no tides at all for most of the time, and would only experience a single tidal nudge every time it passed close to the Sun. If anything, if it locked at all, it should therefore lock to a rotational speed approximately matching its orbital angular velocity near perihelion. But I suspect it just wouldn't lock at all, but would simply retain whatever rotational period it originally started with.
So what's the weather like? Well, for most of its orbit, this planet is going to be so far from the Sun that it might as well be in interstellar space — the Sun will just be the brightest star among many in the sky. Without any significant heat from the Sun, any seas the planet may have will all but freeze solid, and even the atmosphere will freeze and snow down. In general, the surface of such a planet seems pretty inhospitable for life, or at least any kind of life as we know it.
That said, if the planet is large enough, and possesses a massive rocky core, it's possible that the bottom layers of the oceans might stay liquid due to geothermal heat (from trapped primordial heat, and from decay of radioactive elements in the core). Such a subglacial ocean could potentially support life, even if the available free energy would likely be orders of magnitude less than that on Earth's surface.
Thus, we might expect any life on such a planet to be aquatic, and likely to cluster around hydrothermal vents at the bottom of the ocean, like some deep-sea ecosystems do on Earth. Generally, I would not expect life under such conditions to evolve very fast, but in science fiction, even unlikely events are still possible, and so of you want to postulate advanced, even intelligent life to have evolved under such conditions, I'd be happy to suspend my disbelief.
To such an ecosystem, living under a kilometers-thick blanket of ice, safe from to freezing cold of space, would be the norm. The rare and brief passes close to the Sun, if felt at all beneath the thick ice blanket, would be abnormal and catastrophic events: suddenly, scorching rays of heat blast down from what's normally the coldest part of the world, melting the ice and upsetting the ocean circulation.
Of course, any life that did survive such events would eventually adapt to them, but they couldn't possibly be easy on any hypothetical subglacial civilization on your world. Then again, that could make for an excellent story...
